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Solve the given problems by integration.The force $F$ (in $\mathrm{N}$ ) applied by a stamping machine in making a certain computer part is $F=4 x /\left(x^{2}+3 x+2\right),$ where $x$ is the distance (in $\mathrm{cm}$ ) through which the force acts. Find the work done by the force from $x=0$ to $x=0.500 \mathrm{cm}$.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 9

Integration by Partial Fractions: Nonrepeated Linear Factors

Integrals

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equation for work Turn by fools We can write as w is a cool too integration off in isa initial position off X. That is exciting in the final position off every off X, that is, except and if door me. But the quantities are a factor over here. Here we know the d r director we can simplify as the x component off t with unit director I plus why component off tea with unit director G blurs they're component drafty, but you need Director King. Therefore, my work turned w is equal to, as we know, in our new medical, the initial position is evil and the final position is if I change. The if is four x i plus three Reggie into my art director, which is nothing but do you x i plus the off why I so do you off Reggie colors the off zero dickey object most Onley in X direction So vie and their component off the displacement factor are all zero. So in that case, I can dry age integration zero to fi. Now I've enjoyed only the ex combination. So for the force, it is nothing Bird forex, G X and here. There will be the multiplication off I only because we're concerned with the X component only. So we're multiplying this true. And as we know from the scaler rule, I don't. I is equivalent to one. So I'm taking the food as a constant outside off the integration. And we will help zero to fi x ni eggs. Because I don't Isa called you one. No, I'm going to integrate this X And we know the integration off X is X squared upon. Do and eliminate zero Defy. Now, in the next trip, we're going to put the limits. So again, Right four in June, by square upon to insert off X first time, I will put five and instead off X In the second time, I will put the Eagle Square a born true break it complete. And the solution off this whole term will give me the answer. 50 June, which is my work during

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